Question: Solve for $x$, ignoring any extraneous solutions: $\dfrac{x^2 + 33}{x + 7} = \dfrac{x + 89}{x + 7}$
Multiply both sides by $x + 7$ $ \dfrac{x^2 + 33}{x + 7} (x + 7) = \dfrac{x + 89}{x + 7} (x + 7)$ $ x^2 + 33 = x + 89$ Subtract $x + 89$ from both sides: $ x^2 + 33 - (x + 89) = x + 89 - (x + 89)$ $ x^2 + 33 - x - 89 = 0$ $ x^2 - 56 - x = 0$ Factor the expression: $ (x + 7)(x - 8) = 0$ Therefore $x = -7$ or $x = 8$ However, the original expression is undefined when $x = -7$. Therefore, the only solution is $x = 8$.